#6346895

order	512
nilpotency class	3
isoclinism	isoclinic class of #18316
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#62686	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2859 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(17,18,22,25)(19,26,28,31)(20,24,21,29)(23,30,32,27);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(2,8)(7,15)(9,13)(10,16)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

2^9|a^2=i,d^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=dhi,f^a=fh,f^b=fh,g^b=gi,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6254091);